In psychological research sometimes non-parametric procedures are in use in cases, where the corresponding parametric procedure is preferable. This is mainly due to the fact that we pay too much attention to the possible violation of the normality assumption which is needed to derive the exact distribution of the statistic used in the parametric approach.
An example is the t-test and its non-parametric counterpart, the Wilcoxon (Mann-Whitney) test. The Wilcoxon test compares the two distributions and may lead to significance even if the means are equal due to the fact that higher moments in the two populations differ. On the other hand the t-test is so robust against non-normality that there is nearly no need to use the Wilcoxon test. In this paper results of a systematic research of the robustness of statistical procedures against non-normality are presented. These results have been obtained in a research group in Dummerstorf (near Rostock) some years ago and have not been published systematically until now. Most of the results are based on extensive simulation experiments with 10 000 runs each. But there are also some exact mathematically derived results for very extreme deviations from normality (two- and three-point distributions). Generally the results are such that in most practical cases the parametric approach for inferences about means is so robust that it can be recommended in nearly all applications.